That was a perfectly reasonable argument when vaccine supply was limited and there was a huge number of older/weaker people needing vaccines. And that was, in fact, the approach that almost all countries took at the early stages of vaccine rollout.
But it's not a good argument now. At least not in the countries with high uptake. As the vaccines aren't 100% effective, reducing transmissibility will save lives. Still.
And it is absolutely true that the young are much less likely to get severely affected from catching COVID. And that, in a nutshell, is also likely to be the primary reason for lower uptake in the young. It doesn't affect them so they don't really care. It's an entirely rational approach, if somewhat selfish.